let n be Element of NAT ; :: thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-min (L~ (Cage C,n))) .. (Upper_Seq C,n) < (W-max (L~ (Cage C,n))) .. (Upper_Seq C,n)
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); :: thesis: (W-min (L~ (Cage C,n))) .. (Upper_Seq C,n) < (W-max (L~ (Cage C,n))) .. (Upper_Seq C,n)
set Wmi = W-min (L~ (Cage C,n));
set Wma = W-max (L~ (Cage C,n));
set Nmi = N-min (L~ (Cage C,n));
set Nma = N-max (L~ (Cage C,n));
set Ema = E-max (L~ (Cage C,n));
set Rot = Rotate (Cage C,n),(W-min (L~ (Cage C,n)));
A1: Upper_Seq C,n = (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) -: (E-max (L~ (Cage C,n))) by JORDAN1E:def 1;
A2: L~ (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) = L~ (Cage C,n) by REVROT_1:33;
W-min (L~ (Cage C,n)) in rng (Cage C,n) by SPRECT_2:47;
then A3: (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) /. 1 = W-min (L~ (Cage C,n)) by FINSEQ_6:98;
A4: W-min (L~ (Cage C,n)) in rng (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by A2, SPRECT_2:47;
A5: E-max (L~ (Cage C,n)) in rng (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by A2, SPRECT_2:50;
A6: (W-min (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) < (W-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by A2, A3, SPRECT_5:22;
A7: (N-min (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) < (N-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by A2, A3, SPRECT_5:25;
A8: (N-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) <= (E-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by A2, A3, SPRECT_5:26;
A9: (W-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) < (N-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by A2, A3, A7, SPRECT_5:24, XXREAL_0:2;
then (W-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) < (E-max (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by A2, A3, SPRECT_5:26, XXREAL_0:2;
then A10: (W-min (L~ (Cage C,n))) .. ((Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) -: (E-max (L~ (Cage C,n)))) = (W-min (L~ (Cage C,n))) .. (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by A4, A5, A6, SPRECT_5:3, XXREAL_0:2;
W-max (L~ (Cage C,n)) in rng (Rotate (Cage C,n),(W-min (L~ (Cage C,n)))) by A2, SPRECT_2:48;
hence (W-min (L~ (Cage C,n))) .. (Upper_Seq C,n) < (W-max (L~ (Cage C,n))) .. (Upper_Seq C,n) by A1, A5, A6, A8, A9, A10, SPRECT_5:3, XXREAL_0:2; :: thesis: verum